150-152] 



Roche's Model 



149 



forms the boundary intersects itself, so that at such a point the boundary 

 must have the shape either of a sharp edge or of a conical point. 



Let us now examine the various types of problem in detail, beginning 

 with the rotational problem. 



The Rotational Problem 



152. To discuss the problem of a freely-rotating mass, we put F r =0, so 



that 



The condition for a point of equilibrium will obviously first be satisfied in 

 the plane of xy. It will be satisfied at x, 0, if 



MX 



and so is first satisfied when 



where T O is the radius of the cross-section in the plane of xy. The particular 

 equipotential on which this point of equilibrium occurs is found to be 



M 2 I 



T 



Since OT 3 = M/a) z , this equation may be written in the form 



i + ig = f^ (401), 



where sr 2 stands for a? 2 + y 2 . 



The general equipotentials are found to lie as in fig. 28, the critical equi- 

 potential being drawn thick. 



Fig. 28. 



